
Well, duh. All that time I spent trying to debug my Linux From Scratch installation's networking, recompiling the kernel and modules, and even switching over to a crummier old D-Link NIC was entirely unnecessary.
There's a little file that lives in /etc called resolv.conf, that's kind of necessary if you expect your computer to be able to resolve any domain names that don't happen to be contained in /etc/hosts. I have dealt with this file before, but for whatever reason it didn't occur to me that my problem might be DNS related this time, and so I forgot about it.
Sure enough, I created my /etc/resolv.conf file and I'm out on the internet. I'm typing this entry in Lynx right now, actually.
This just goes to show that it's always a good idea to learn things the complicated way even if you wind up using a tool like linuxconf or netconf in the long run; it really pays to know where all the little config files live, even though it's easier to fill them all out in a configuration tool.